Solution: Rewrite as $x^2 = 4y^2 + 100$. Let $x = 2k$, then $4k^2 = 4y^2 + 100 \implies k^2 - y^2 = 25$. Factor as $(k - y)(k + y) = 25$. The factor pairs of 25 are $(1,25), (5,5), (-1,-25), (-5,-5)$. Solving:

["Title: Solving the Equation $x^2 = 4y^2 + 100$ Using Algebraic Substitution and Factoring", "When working with quadratic equations, especially those resembling the difference of squares, a clever substitution and factoring strategy can simplify the process significantly. Consider the equation:", "$$\nx^2 = 4y^2 + 100\n$$", "This equation models a hyperbolic relationship between $x$ and $y$, but it can be rewritten in a cleaner, solvable form.", "---", "### Step 1: Rearranging the Equation", "Begin by isolating terms on one side:", "$$\nx^2 - 4y^2 = 100\n$$", "This is a difference of squares: $x^2 - (2y)^2 = 100$. Apply the identity:", "$$\na^2 - b^2 = (a - b)(a + b)\n$$", "Let $a = x$ and $b = 2y$. Then:", "$$\nx^2 - (2y)^2 = (x - 2y)(x + 2y) = 100\n$$", "---", "### Step 2: Substitute $x = 2k$ to Simplify", "Let $x = 2k$. Substituting into the factored form:", "$$\n(2k - 2y)(2k + 2y) = 100\n$$", "Factor out the 2s:", "$$\n2(k - y) \cdot 2(k + y) = 4(k - y)(k + y) = 100\n$$", "Divide both sides by 4:", "$$\n(k - y)(k + y) = 25\n$$", "This transformation reduces the original nonlinear equation into a simpler symmetric form:\n$k^2 - y^2 = 25$, which arises naturally from the substitution.", "---", "### Step 3: Use Factor Pairs of 25 to Solve", "We now solve $(k - y)(k + y) = 25$. Since 25 has integer factor pairs $(1,25), (5,5), (-1,-25), (-5,-5)$, consider each:", "1. $k - y = 1$, $k + y = 25$\n Add: $2k = 26 \implies k = 13$\n Subtract: $2y = 24 \implies y = 12$", "2. $k - y = 5$, $k + y = 5$\n Add: $2k = 10 \implies k = 5$\n Subtract: $2y = 0 \implies y = 0$", "3. $k - y = -1$, $k + y = -25$\n Add: $2k = -26 \implies k = -13$\n Subtract: $2y = -24 \implies y = -12$", "4. $k - y = -5$, $k + y = -5$\n Add: $2k = -10 \implies k = -5$\n Subtract: $2y = 0 \implies y = 0$", "---", "### Step 4: Recover $x$ Using $x = 2k$", "From each $(k, y)$ solution:", "- $k = 13 \Rightarrow x = 26$, $y = 12$\n- $k = 5 \Rightarrow x = 10$, $y = 0$\n- $k = -13 \Rightarrow x = -26$, $y = -12$\n- $k = -5 \Rightarrow x = -10$, $y = 0$", "All valid pairs:\n$$\n(x, y) = (\pm26, \pm12),\quad (\pm10, 0)\n$$", "---", "### Summary", "By rewriting $x^2 = 4y^2 + 100$ as $(x - 2y)(x + 2y) = 100$, then substituting $x = 2k$, we reduced the problem to solving a difference of squares via integer factor pairs. This method offers a structured, algebraic path from a nonlinear equation to its exact solutions — ideal for both learning and algorithmic problem-solving.", "---", "This technique applies broadly to equations of the form $x^2 - ky^2 = \ ext{constant}$, and reinforces how substitution and factoring simplify seemingly complex problems. Whether solving for real roots or preparing for graphing, understanding the structure $k^2 - y^2 = 25$ reveals the elegant symmetry hidden within.", "---", "Keywords: solve $x^2 = 4y^2 + 100$, factor $x^2 - 4y^2 = 100$, difference of squares, substitution $x = 2k$, factor pairs, solve quadratic equations, algebra trick, coordinate geometry, hyperbola equation solver"]